For any four points on a plane, if the areas of four triangles formed are different positive integer and six distances between those four points are also six different positive integers, then the convex closure of $4$ points is called a "lotus design." (1) Construct an example of "lotus design". Also what are areas and distances in your example? (2) Prove that there are infinitely many "lotus design" which are not similar.
Problem
Source: China South East Mathematical Olympiad Grade 10 Prob. 4
Tags: combinatorics
FC_YangGuifei
29.05.2018 11:53
Any Solution?
MarkBcc168
29.05.2018 13:58
Idea : Take isosceles trapezoid. The rest is straightforward length bashing (which I'm too lazy to recreate it here.)
ThE-dArK-lOrD
29.05.2018 19:24
@above Isosceles trapezoid not yields those different conditions in the problems? For this problem, there're two construction I know:
Use two right triangles sharing the same hypotenuse with proper side lengths, the resulted quadrilateral will be concyclic and Ptolemy's gives the sixth side.
Consider triangle $ABC$ with orthocenter $H$, reflect $H$ over $BC$ to get $H'$, then properly choose side lengths of $ABC$ (one can use Pell's to create 13-14-15-like triangle.)
MACGN
21.01.2023 10:42
The "lotus design" should be essentially called as "Heron quadrilateral",because its background is how to creating (or meaning composing) a Heron triangle